To calculate the square root of any number, estimate by using the closest perfect square, and divide the number for which you want the square root by the root of that perfect square. Average the result with that perfect square root, repeating the process as you approach the closest number possible.
- Find the closest perfect square to your number
Look at a list of perfect squares, and find the one closest to your number. Consider the task of finding the square root of 15; 16 is the closest perfect square, as its square root (4) is also an integer.
- Divide and average your number by the square root of that perfect square
Divide 15 by 4 for the purposes of this example, and write down the answer (3.75). Take the average of the answer and 4, which is 3.875. This is now your closest estimate for the square root of 15. Use this closest estimate for future averaging steps in this problem.
- Divide and average to find the exact number
Keep dividing and averaging until you find the answer that multiplies by itself to get as close to 15 as possible. Divide 15 by 3.875 for the purposes of this example, and write down the answer (3.871). Average the answer with 3.875 to yield 3.873. Multiply 3.873 by itself to see how close you are, and decide if the answer (15.000129) is close enough. Expand the number of decimal places you are willing to go with your estimate to gain greater precision in your final answer.